Spectral data of special orthogonal Higgs bundles and Hecke modification

ABSTRACT We give a complete, self-contained computation of the spectral data parametrizing Higgs bundles in the generic fibres of $\mathrm{SO}_{2n+1}$-Hitchin fibration where the Higgs fields are L-twisted endomorphisms. Although the spectral data are known in the literature, we develop a new approach, which takes advantage of Hecke modification. Furthermore, we present the computation for $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n}$ cases while clarifying some aspects of the correspondence, which are not well explained in the pre-existing literature. We also compute the number of connected components of the generic fibres and demonstrate Langlands duality in the fibres via the canonical duality in the fibres.